Harro Schmeling

Scaling of Newtonian and non-Newtonian fluid dynamics without inertia for quantitative modelling of rock flow due to gravity (including the concept of rheological similarity)

Scale model theory for constructing dynamically scaled analogue models of rock flowing in the solid state has until now assumed that the natural and model flows were both viscous. In viscous flows, at the very low Reynolds numbers (Re ⪡ 1) common in solid rocks, geometrical similarity is sufficient to achieve dynamic similarity between a homogeneous material (scale) model and its natural prototype. However, experiments on the rheology of natural rocks suggest that they flow predominantly as non-Newtonian strain rate softening materials at the characteristic geological strain rate 10−14 s−1. Non-dimensionalisation of both the equation of motion and the constitutive flow law of non-Newtonian flows is carried out to investigate what criteria are required to achieve dynamic similarity. It is shown that dynamic similarity of non-Newtonian flows at low inertia (e.g., a rock with Re ⪡ 1 and its model analogue) can only be attained if the steady-state flow curves of the model materials and the various rocks in the prototype have mutually similar shapes and slopes, and if these flows operate on similar parts of their respective flow curves. We term this the requirement of rheological similarity. Dynamic similarity of low inertia flows (Re ⪡ 1) in non-Newtonian continua is achieved if they are rheologically and geometrically similar. Additional criteria for dynamic similarity of low inertia flows in inhomogeneous media (with Newtonian or non-Newtonian subregions, or both) are formulated in section 5. Scaling of thermal properties is not included. Steady-state flow curves of common rocks are compiled in log stress-log strain rate space together with analogue materials suitable for modelling of solid state rock deformation. This compilation aids the selection of model materials with flow curves geometrically similar to those of rocks in the prototype. Laboratory scale models of rock flow should generally be constructed of materials which strain rate soften during flow at the convenient shear rate 10−2 s−1.

A comparison of methods for the modeling of thermochemical convection

We have compared several methods of studying thermochemical convection in a Boussinesq fluid at infinite Prandtl number. For the representation of chemical heterogeneity tracer, marker chain, and field methods are employed. In the case of an isothermal Rayleigh-Taylor instability, good agreement is found for the initial rise of the unstable lower layer; however, the timing and location of the later smaller-scale instabilities may differ between methods. For a simulation of entrainment by thermal convection of a dense layer at the bottom of the mantle we found good agreement for a few overturn times. After this, differences between the results can be large. We propose intrinsic differences between the methods and possibly chaotic mixing effects may be the cause of the lack of detailed agreement. The comparison shows that high resolution is necessary for a reasonable thermochemical study. This will pose severe restrictions on the applicability of these methods to three-dimensional situations.

Delamination and detachment of a lithospheric root

Abstract Orogens can be formed by shortening of the lithosphere on time scales much shorter than that of thermal diffusion. For time scales of a few Ma, the major part of the uplift of an orogen, which may occur during the transition from crustal shortening to extension, may be due to delamination of the mantle lithosphere rather than to convective thinning. The negative buoyancy forces of a lithospheric root may cause lateral shortening, while the lithospheric root delaminates (peels off) from the crust, detaches, and is replaced by hot upwelling asthenosphere. This provides a large input of heat into the crust. Geological observations show that the late orogenic phase may be characterized by volcanic activity, high temperature–low pressure metamorphism, extension, high heat flow, and a rapid (within a few Ma) increase of the topography. We present dynamically and thermally consistent numerical models of the evolution of an orogenic root. Mantle rheology is based on laboratory data of olivine. While models with the rheology of olivine do not allow for delamination due to its very high viscosity, the incorporation of a quasi brittle rheology is necessary for modelling delamination of the mantle lithosphere. Successful delamination models are always associated with a rapid change in topography. We examined the parameter range of our models and found different regimes of behaviour of the orogenic root of the mantle lithosphere: no delamination, delamination, delamination and full detachment. For a fixed viscosity of the upper crust of 1023 Pa s delamination occurs for a range of viscosities of 1023 Pa s to 1020 Pa s for the lower crust, depending on the thickness of the lithospheric root. But full detachment of the delaminated lithospheric slab only occurs if the viscosity of the lower crust is greater than approximately 1021 Pa s. Hot asthenosphere is welling up into the gap opening during delamination. Further implications of our models are: Lithospheric roots or unsupported slabs of at least 100–170 km depth extent are needed to provide sufficient negative buoyancy to allow delamination and detachment. In all delamination models significant amounts of lower crustal material are subducted into the mantle. In applying our models to selected orogens we argue that mantle delamination may have played an important role in the Variscan and the Himalayan orogenies.

Numerical models on the influence of partial melt on elastic, anelastic and electric properties of rocks. Part I: elasticity and anelasticity

Abstract With the aim of a simultaneous interpretation of elastic, anelastic and electric in situ data from the asthenosphere a comprehensive set of numerical models is developed for partial melt in different geometrical configurations. For the elastic and anelastic modulus use is made throughout of the melt squirt mechanism. Frequency dependence is not treated in detail but estimated from the limiting cases of the relaxed and unrelaxed modulus. This has the advantage that quantitative values of viscocity and flow path dimensions are not required. In the models melt can be assumed to occur in the form of tubes, films, and triaxial ellipsoidal inclusions of arbitrary aspect ratio. The conditions in which the solutions for triaxial ellipsoidal inclusions can be approximated by simpler ones for spheroidal inclusions are discussed. It is then shown up to which aspect ratio a published model on melt films is applicable. The problem of interconnection of inclusions is treated with a statistical numerical approach. It is found that a reduced degree of interconnection may have a significant influence on anelastic relaxation at melt fractions corresponding to a moderate modulus decrease. A useful representation of the anelastic melt models is introduced by plotting the relaxation strength against the effective modulus, both of which depend on the state of melting. Such diagrams allow a clear distinction between the different melt geometries and may be used for the interpretation of observed data. Finally, different melt geometries are superimposed and it is found that under certain conditions bulk dissipation may reach the order of that for shear.

Shear modulus and Q of forsterite and dunite near partial melting from forced-oscillation experiments

An apparatus designed to determine the complex shear modulus of rock samples by forced torsion oscillations at high temperature and in the seismic frequency band 0.003–30 Hz is briefly described. Measurements were performed on natural dunite from Aheim (Norway) up to 1400°C and on polycrystalline forsterite up to 1500°C at 1 atm pressure. The two materials were chosen to study, by comparison, the effect of melt on the elasticity and anelasticity of mantle rocks. Between 1000 and 1200°C the absolute values of the shear modulus G are almost equal for both materials. Above 1200°C G for natural dunite decreases progressively with temperature and at 1400°C and 1 Hz reaches ∼13 of its value at 1100°C. In contrast, G of pure forsterite depends little on temperature. For petrological reasons, supported by simultaneous measurements of the electric resistivity, there is strong evidence that the decrease of G in dunite above 1200°C is due to melt from the lower melting components of the dunite. Based on different models estimates of the melt fraction are made. At high temperature, in both materials Q−1 is characterized by a monotonic decrease with frequency according to ω−α, with α ≈ 0.25. An apparent activation energy of 38±5 kcal mol−1 for forsterite and 48±8 kcal mol−1 for dunite was found with no significant change in the regime of partial melting. From this it is concluded that Q−1, even at partial melting, is dominated by solid state high temperature background absorption. There is no indication from these experiments for a constant-Q-band at low seismic frequencies or an increase of Q proportional to frequency as suggested by some seismologists. The present results are in good qualitative agreement with those for Youngs modulus obtained previously by strain retardation experiments.

Large-scale lithospheric stress field and topography induced by global mantle circulation

Stresses in the lithosphere are one indication of processes in the Earth interior: here we present a calculation of largescale lithospheric stresses caused by global mantle circulation. The mantle flow field is calculated based on density structures inferred from global seismic tomography. Predicted principal stress directions are compared to interpolations based on observed stresses. Agreement between predictions and observations is often good in regions where lithospheric stresses and mantle tomography are well constrained. Predicted magnitudes of scalar stress anomalies vary more strongly than predicted stress directions for various tomographic models. Hotspots preferentially occur in regions where calculated stress anomalies are tensile or slightly compressive. Results do not strongly depend on radial mantle viscosity structure, lithospheric rheology (viscous or elastic) or plate motion model. The model also predicts the directions of motion well for most plates; misfits in the predicted magnitudes can be explained qualitatively. Stress anomalies due to causes within the lithosphere (oceanic cooling with age, variations in crustal thickness, topography isostatically compensated at subcrustal levels) are also computed. Predicted stress directions in the absence of mantle flow can explain observations almost as well as mantle flow. Nevertheless, current models of mantle flow are largely in accord with interpolations of observed principal stress directions and the observed plate motions. fl 2001 Elsevier Science B.V. All rights reserved.

Numerical models on the influence of partial melt on elastic, anelastic and electrical properties of rocks. Part II: electrical conductivity

Abstract Partial melting influences both the elastic and electrical properties of mantle rocks significantly, but in a different manner. A set of theoretical models of the electrical conductivity of partially molten rocks is developed for different geometries of the melt distribution. The models correspond to a similar set on elasticity and anelasticity presented in a preceding paper (part I). Both model sets allow combined interpretations of seismic and geoelectric observations from anomalous hot mantle regions by a systematic variation of the melt geometry. The following melt geometries are assumed in the conductivity models: melt films, tubes, isolated ellipsoidal inclusions, films and ellipsoidal inclusions with a variable degree of interconnection, distributed geometries, and superpositions of spheres with films and with tubes. Unexplored conduction mechanisms near the melting temperature, film thicknesses of molecular sizes, or effects of volatiles are not considered. Resistor network theory is utilized to account for a variable degree of interconnection and distributed geometries. Thereby, a new possible interpretation of Archies law is found in terms of varying interconnection. Furthermore, it is found that a reduced degree of interconnection may have an important influence on the conductivity at melt fractions characteristic for a weak to moderate decrease in seismic velocity. The model results are compiled together with the elastic models of part I in a set of diagrams which may be applied systematically to observed electrical and seismic data to find constraints on the state of melting.

Finite deformation in and around a fluid sphere moving through a viscous medium: implications for diapiric ascent

Abstract Different types of geodynamically active regions like mountain belts, hot spots, or areas of salt tectonics are characterized by diapirism. One key for the reconstruction of the dynamic history of such structures is the progressive strain, which can sometimes be determined from field observations. Successful interpretation of such observations requires a quantitative model of the finite strain in and around a rising diapir. A constant viscosity fluid sphere of radius R rising through another isoviscous fluid is assumed to approximate the buoyant motion of a diapir. Known analytical solutions for the velocity fields are used to numerically evaluate the finite deformation in and around the sphere. Regions of very high strains are found in a tube with a radius 1 2 R behind the sphere and in a shell of thickness of 0.1R around the sphere. The following three-dimensional strain regimes can be identified: In the half space above the sphere down to its equator finite strains are oblate. Behind the sphere they progressively change to plane strain. In the diapiric source region they are prolate. Viscous drag at the spheres surface leads to an internal circulation with one overturn after 10R of rise, if the sphere has a relatively low viscosity. Finite strains within the fluid sphere show a continuous increase with superimposed cyclic straining and unstraining. After several body radii of rise, the strains become highly inhomogeneous inside the sphere except along the vertical axis and just inside the spheres surface, where strong prolate and oblate strains are observed, respectively. Finite strain determinations in a falling ball experiment (Cruden, 1988) are compared with the theoretical results. At horizontal distances of a few body radii from the fall axis, the effect of confining container walls is clearly seen in the experimental strain data. The results are compared briefly with available strain data from the field which seem to be significantly lower than predicted. Proposed explanations include a short memory of the rock fabric, and a lack of recognition of the strain concentration which would be expected for a temperature-dependent rheology. It might also be possible that few natural diapirs rise more than a few radii in the solid state.

Polydiapirs: multiwavelength gravity structures

Polydiapirs (domes-in-domes) result from the evolution of a multiwavelength Rayleigh-Taylor (gravitational) instability. The present work investigates the initiation and development of polydiapirs by means of two-dimensional finite-difference models of gravitationally unstable triple-layered sequences of Newtonian fluids. The values of viscosities, thicknesses and densities were systematically varied for selected examples. In triple-layered sequences, only two types of density stratification are likely to result in multiwavelength instabilities that will eventually develop into polydiapiric structures. The limiting factors and evolution of such structures were studied for both these types of density stratifications. Finite-difference calculations on models with physical parameters appropriate to evaporitic sequences and covered by a viscous clastic overburden showed that a very small downward density decrease in the evaporitic source may cause an internal overturn that will compete with the main overturn between source and overburden. This may lead to the development of sequential or simultaneous polydiapirs. The mature polydiapirs in the models exhibit spines, curtain folds, pendant repetitions of the source layers inside the large diapirs, and anomalous interdiapiric distances. All these features are known in natural salt diapirs suggesting that polydiapirs may be far more common in nature than hitherto recognized.

Partial Melting and Melt Segregation in a Convecting Mantle

Various causes for mantle melting (decompression, heating or release of water) combined with current estimates of upper mantle temperatures and the state of stress in the lithosphere suggest that in many regions the asthenosphere might be partially molten, but melts may not always be able to rise to the surface. The governing equations describing melting, melt segregation, compaction and depletion in a deforming medium are discussed with emphasis on the physical processes involved. To combine these processes with a convecting upper mantle flow, a “Compaction Boussinesq Approximation” (CBA) is introduced and tested with known solutions. Driving forces include thermal, melt, depletion and enrichment buoyancy. The bulk viscosity and its dependence on porosity has a significant effect on the melt flow even for distances large compared to the compaction length. 1D and 2D solitary porosity waves are discussed with particular emphasis on a variable bulk viscosity, compaction, and dilatation of the matrix. Melting, segregation and solidification processes are studied in a self-consistent model of a variable viscosity plume head arriving at the base of the lithosphere. It is shown that melt buoyancy dominates segregation velocities. However, a variable bulk viscosity may still have some influence on the segregation velocities, while dynamic pressures may be neglected. In the absence of a mantle plume a partially molten undepleted asthenosphere may develop melting instabilities, driven by thermal, melt and depletion buoyancy. This instability propagates laterally with velocities of the order of several cm/a and has a length scale of about 2 times the thickness of the partially molten asthenosphere. Volcanism associated with this propagating instability might have a similar appearance as hot spot tracks suggesting that this instability might be an alternative mechanism to the plume hypothesis at least for some volcanic chains.